Symmetries of projective spaces and spheres

TL;DR

This paper characterizes angle-preserving bijections on projective spaces over real and complex inner product spaces, extending Wigner's theorem in quantum mechanics without assuming completeness, and explores their structure for specific dimensions and angles.

Contribution

It provides a new characterization of angle-preserving transformations on projective spaces, generalizing Wigner's theorem without the need for the space to be complete.

Findings

Characterization of bijections preserving a fixed angle in projective spaces.

Extension of Uhlhorn's theorem to non-complete inner product spaces.

Identification of additional transformations in the case of two-dimensional complex spaces at  = 4

Abstract

Let $H$ be either a complex inner product space of dimension at least two, or a real inner product space of dimension at least three. Let us fix an $\alpha\in \left(0,\tfrac{\pi}{2}\right)$. The purpose of this paper is to characterize all bijective transformations on the projective space $P(H)$ obtained from $H$ which preserves the angle $\alpha$ between lines in both directions. (We emphasize that we do not assume anything about other angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$, and when $H$ is a complex inner product space of dimension at least three we describe the structure of these transformations for $\alpha\leq\tfrac{\pi}{4}$. As an application, we give an Uhlhorn-type generalization of a famous theorem of Wigner which is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, we show that under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos^2\alpha$ in both directions is a Wigner symmetry (i.e. it automatically preserves all transition probability), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi}{4}$ where an additional possibility occurs. We note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi}{2}$ case. Usually in the literature, results which are connected to Wigner's theorem are discussed under the assumption of completeness of $H$, however, here we shall remove this unnecessary hypothesis. Our main tools are a characterization of bijective maps on unit spheres of real inner product spaces which preserve an angle in both directions, and an extension of Uhlhorn's theorem for non-complete inner product spaces.